Figure 1

2D panels: phase diagram showing the number of stable solutions as a function of the rescaled pump intensities $F_{1,2}^{\prime }=\sqrt{g_x}{F}_{1,2}$ meV${}^{3/2}$. White, green, yellow, and black regions correspond to zero, one, two, or three stable solutions, respectively. In panels I and II, ${\mathbf{k}}_1=0.25$ $\mu$m${}^{-1}$ and ${\mathbf{k}}_2=0.7$ $\mu$m${}^{-1}$ while in panels III and IV, ${\mathbf{k}}_1=0.0$ $\mu$m${}^{-1}$ and ${\mathbf{k}}_2=0.7$ $\mu$m${}^{-1}$. In the left panels (I and III), ${\omega }_{1,2}={\omega }_{\text{LP}}({\mathbf{k}}_{1,2})+0.3$ meV, while in the right panels (II and IV), ${\omega }_{1,2}={\omega }_{\text{LP}}({\mathbf{k}}_{1,2})+0.4$ meV. The horizontal black line lies at the three fixed values of $F_2^{\prime }$ corresponding to the three panels of Fig. 2, while the blue diagonal line is the path used to plot Fig. 5. 3D panels: plots of $g_X\left|{\psi }_X^{\mathit{ss}}\right|{}^2$ in meV as a function of $F_{1,2}^{\prime }$ with parameters equal to panel I. Stable solutions with higher populations are shown in green, stable solution with the second higher population in yellow, and third stable solution with lower population in black. All the solutions are shown in the left panel. Since the upper green branches hide a yellow upper branch, the right panel shows only the yellow and black solutions.Reuse & Permissions

Figure 2

Stability curves of the total exciton density $g_X\left|{\psi }_X^{\mathit{ss}}\right|{}^2$ in meV (red dotted curves represent unstable solutions, black lines are stable solutions) for fixed pump intensities as a function of $F_1^{\prime }$ for $F_2^{\prime }=0.00001$ meV${}^{3/2}$ (top left), $F_2^{\prime }=0.025$${\mathrm{meV}}^{3/2}$ (bottom left), and $F_2^{\prime }=0.08$ meV${}^{3/2}$ (right). Points $A_i$ ($B_i$), $i=1,2,3$ correspond to the cases when the lower (upper) branch of the stability curve becomes unstable (see Figs. 3 and 4). Point $C$ does not have a counterpart in the one-fluid case and corresponds to the cases when the second high branch of the stability curve becomes unstable (see Fig. 4).Reuse & Permissions

Figure 3

Dispersion of the imaginary part of the excitation eigenfrequency $\omega ={\text{LP}}_j^{\pm }$. The three panels correspond to points $A_i$ with $i=1,2$, and $3$ of Fig. 2, where the lower part of the stability curves become unstable. In blue (red) the parts corresponding to the scattering of two particles with $\mathbf{k}={\mathbf{k}}_1$ ($\mathbf{k}={\mathbf{k}}_2$).Reuse & Permissions

Figure 4

Dispersion of the imaginary part of the excitation eigenfrequency $\omega ={\text{LP}}_j^{\pm }$. The three panels correspond to points ${\mathrm{B}}_i$ with $i=1,2,3$ and C of Fig. 2, where the higher part of the stability curve becomes unstable.Reuse & Permissions

Figure 5

Hysteresis cycles of ${\alpha }_1=\left|{\psi }_{1_X}^{\mathit{ss}}\right|{}^2/(|{\psi }_{1_X}^{\mathit{ss}}|{}^2+|{\psi }_{2_X}^{\mathit{ss}}|{}^2)$ (dimensionless) as a function of $F_1^{\prime }$ for different values of $F_2^{\prime }$—stable solutions are in black, while the hysteresis cycle performed by the system is in brown. (a): $F_2^{\prime }=0.14-F_1^{\prime }$ as in the blue line of Fig. 1. (b): $F_2^{\prime }=0.08$ meV${}^{3/2}$ as in the horizontal black line of Fig. 1.Reuse & Permissions